Estimation of factor loadings

The estimation results for the fully specified Kalman filter based model in (7.18)–

(7.19) are provided in Table 7.6. The estimated hyperparameters are broadly sig-nificant. While the market factor is significant for all sectors except for Travel &

Leisure, both fundamental factors are significant at the 1% level for all sectors.

With regard to the set of macroeconomic factors it can be observed that not all factors are relevant for all sectors. The term structure factor, which is supposed to pick up systematic risks that are related to a changing slope of the yield curve, turns out to be significant for thirteen out of eighteen sectors. The estimated sen-sitivities to changes in the oil price are significantly different from zero for twelve sectors. Surprisingly, the oil price does not represent a systematic risk factor for the energy-related Chemicals, Utilities and Travel & Leisure sectors. This leaves the latter as the only sector with only four significant factors. The overall most important macroeconomic factor in explaining the sector return series over time is the exchange rate. With the exception of Healthcare, Personal & Household Goods and Telecommunications, the dollar is significant for all sectors. As measured by the adjusted coefficient of determination, the proposed specification with the selected set of explanatory variables explains between 65% and 93% of the variance of sec-tors. The lowest values of ¯R² are observed for Travel & Leisure and the defensive

sectors Food & Beverages, Healthcare, Oil & Gas and Utilities. The model is found to be more appropriate for financials and non-financial cyclicals than for defensives and TMT.

Regarding the estimation results of the alternative least squares based multiple factor specifications given by (7.20), the finding that the market and fundamen-tal factors are more significant than the macroeconomic factors is only partially maintained. For the RLS specification, both the market and the size factor are sig-nificant for all eighteen sectors. With the exception of Construction and Healthcare, the value-growth spread is significant at least at the 10% level for all sectors. In contrast to the observation made above, the term structure factor also represents a systematic risk factor for sixteen sectors. The exchange rate factor is only signifi-cantly different from zero for Automobiles. Changes in the oil price represent the second most important macroeconomic factor. As expected the explanatory power is lower than in case of the more flexible Kalman filter based model: as measured by the adjusted R² between 49% (Food & Beverages) and 78% (Banks) of the sector variance can be explained by movements of the chosen systematic risk factors. The estimation results for the rolling RR1 and RR5 specifications are summarized by reporting the respective ranges of estimated coefficients. It can be seen that all factors are significant for each sector at some stage. This supports the hypothesis of time-varying factor sensitivities. For a detailed summary of the estimated parame-ters of the alternative least squares based multifactor models, see Table C.7 in the appendix.

Examples of the paths of Kalman filter based factor loadings are illustrated in the following set of figures. All charts show filtered state estimates so that large outliers may occur at the beginning of the respective series of factor loadings. The two panels in Figure 7.1 display the respective time-varying exposure of Technology and Food & Beverages to the value-growth spread. Technology’s value-growth beta became clearly negative during the bubble and its subsequent aftermath, which reflects the growth characteristics of the sector. As can be seen from Panel (b), the market perception of whether the Food & Beverages sector has more growth or more value attributes changes over time. Four distinct regimes can be identified.

Until October 1997, the V GR beta of the sector moves around zero. Over the subsequent two years, when the whole market rose and growth outperformed value, the beta becomes slightly negative. After the market peaked in March 2000, the V GR beta of Food & Beverages turns positive. This is intuitive given the sector’s defensive characteristics: market participants exited the overcrowded and expensive growth segments of the market and looked for cheaper and safer alternatives such as Food & Beverages. This led to a rising V GR beta. At the end of the sample, the estimated sensitivity returns toward zero, which means that the value-growth spread no longer represents a systematic risk factor of the sector. The observation that theV GRbeta is only significantly different from zero around the TMT bubble holds for other sectors as well.

filterbasedconditionalmultifactorpricingmodel147 Table 7.6: Parameter estimates for multifactor Kalman filter models.

This table reports the estimated parameters for the Kalman filter based multifactor specification for the eighteen DJ Stoxx sectors. Columns 3–8 report the estimated state disturbance variance termsσ_ηk² fork= 1, . . . ,6, with kreferring to the respective regressor (1: BMR, 2: VGS, 3: SIZ, 4: TS, 5: OIL, 6: FX); *** means significance at the 1% level (**: 5%, *: 10%). The last column shows adjusted coefficients of determination.

Sector σ²×10⁴ σ²_η1×10⁴ σ_η²2×10⁴ σ_η²3×10⁴ σ²_η4×10⁴ σ_η²5×10⁴ σ²_η6×10⁴ logL BIC R¯² Automobiles 3.14*** 8.80*** 43.37*** 19.61*** 3.03*** 0.11*** 42.13*** 1692.1 −4.89 0.77 Banks 0.78*** 1.68*** 29.82*** 96.09*** 2.63*** 0.00 3.32*** 2105.7 −6.10 0.93 Basics 3.03*** 6.07*** 68.37*** 8.24*** 1.66*** 0.10*** 25.96*** 1716.0 −4.96 0.70 Chemicals 1.92*** 2.52*** 16.46*** 36.45*** 0.90*** 0.00 60.08*** 1861.0 −5.38 0.77 Construction 1.32*** 2.31*** 18.37*** 24.25*** 1.06*** 0.13*** 27.40*** 1979.0 −5.73 0.81 Financials 1.04*** 1.11*** 11.78*** 59.83*** 2.36*** 1.12*** 13.26*** 2025.7 −5.86 0.89 Food 1.84*** 5.66*** 25.15*** 18.80*** 0.22*** 0.01* 6.08*** 1900.2 −5.50 0.65 Healthcare 2.57*** 0.79*** 61.77*** 39.41*** 0.00 0.03*** 0.00 1802.7 −5.21 0.65 Industrials 0.79*** 2.59*** 43.62*** 29.01*** 1.52*** 0.00** 24.84*** 2113.8 −6.12 0.91 Insurance 1.79*** 31.85*** 0.00*** 85.91*** 3.87*** 0.00 8.54*** 1855.6 −5.37 0.89 Media 2.42*** 22.02*** 64.54*** 30.86*** 1.64*** 0.56*** 5.13*** 1775.1 −5.13 0.85 Oil & Gas 2.86*** 0.40*** 12.48*** 48.60*** 0.00 0.91*** 31.06*** 1758.9 −5.08 0.67 Personal 1.53*** 6.73*** 46.08*** 19.53*** 0.71*** 0.00 0.00 1946.7 −5.63 0.79 Retail 1.69*** 2.62*** 9.74*** 31.96*** 0.22*** 0.20*** 16.79*** 1923.5 −5.57 0.72 Technology 3.04*** 36.69*** 165.38*** 28.17*** 4.10*** 0.12*** 11.59*** 1680.2 −4.85 0.88 Telecom 3.21*** 2.99*** 22.99*** 20.19*** 0.00 0.03*** 0.00 1738.8 −5.02 0.78

Travel 3.14*** 0.00 1.56*** 47.01*** 0.00 0.00 43.06*** 1737.6 −5.02 0.66

Utilities 1.52*** 4.65*** 21.57*** 29.40*** 0.00 0.00 4.10*** 1970.7 −5.70 0.69

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Figure 7.1: V GR factor loadings for Technology and Food & Beverages.

Figure 7.2 displays the time-varying size exposure for Industrials and Healthcare.

While the Industrials sector index is comprised of many small caps, the Healthcare sector is dominated by large cap pharmaceutical stocks, such as GlaxoSmithKline, Novartis and Sanofi-Aventis. Therefore, size beta of Healthcare can be expected to be higher. At the first glance, both series mostly move in positive territory, which is representative for all sectors. This means that over the entire estimation period returns are usually positively related to an outperformance of large caps over small caps. As can be seen from Panel (a), the size beta of the Industrials sector occasionally becomes negative and rarely takes on values greater than 0.5. This is in contrast to the average size beta of Healthcare, which moves around unity as shown in Panel (b).

Figure 7.2: SIZ factor loadings for Industrials and Healthcare.

Changes in the slope of the yield curve are generally assumed to capture changes in the state of the economy and would naturally be expected to have a systematic impact on cyclical sectors. The two panels in Figure 7.3 illustrate the paths of T S loadings for the non-financial cyclical sector Automobiles and the financial cyclical

Insurance sector, respectively. For both sectors, the sensitivities alternate between positive and negative territory: at some stages of the cycle the corresponding sectors rise when the yield curve steepens, while at other times the returns are positively related to a flattening of the curve.

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Figure 7.3: T S factor loadings for Automobiles and Insurance.

For the Oil & Gas sector, the loadings on the oil price are positive over the complete out-of-sample period (t=163,. . . ,683). This could be expected as the sales and earnings of oil companies are directly linked to changes in the price of oil. This does not mean that the systematic importance of the oil price factor is not changing over time. As illustrated by the left panel of Figure 7.4, theOIL beta of the sector even decreases toward zero after July 2001. At the same time, the sector’s V GR beta starts to pick up. Between 2002 and 2003 the value-growth spread becomes relatively more important in explaining the series of Oil & Gas returns than changes in the price of oil.

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Figure 7.4: OIL and V GR betas for the Oil & Gas sector.

As indicated above, the US-dollar is found to be the most important systematic macroeconomic risk factor. All sectors tend to be negatively related to a rising euro.

This observation is especially true for export-oriented industrial cyclical sectors such as Industrials and Automobiles; it only holds to a lower degree for Utilities as shown by Figure 7.5.

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−0.2 0.0 0.2 0.4

(a) Utilities

βFX

1994 1998 2002 −1.5

−1.0

−0.5 0.0

(b) Industrials

βFX

1994 1998 2002

Figure 7.5: F X factor loadings for Utilities and Industrials.